The variance of the ratio of two normals does not exist if the
denominator has a mean of zero. Then yes we get a Cauchy distribution
with all the ugly properties. If the mean is not zero, you are not
dividing by zero very often, although I don't really know whether that
distribution of that ratio is generally known.
Back to Sergiy's question -- to gauge the differences between the
variance estimation methods, you can also try -svy, jackknife- and see
what it produces; the differences on -sysuse auto- dataset should be
somewhere in the fourth or fifth decimal point. But I think Austin
nailed down the analytical problem with an extra `rho'.
Of course you are shooting sparrows with a cannon -- you probably
could've achieved anything you needed using -nlcom-.
On 1/13/09, Feiveson, Alan H. (JSC-SK311) <[email protected]> wrote:
> It should also be noted that this delta method is just an approximation
> - so it is not surprising that it might disagree with a simulaiton by
> 10% or more. Also, technically the variance of a ratio of two normally
> distributed random variables doesn't even exist! Therefore even a
> simulation, if carried out long enough would produce arbitarily high
> "SE" values. The saving grace is that if the variance of the denominator
> is much smaller than the mean, we can "get away" with these
> approximations for practical usage.
>
> Al Feiveson
>
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Jeph Herrin
> Sent: Tuesday, January 13, 2009 7:11 AM
> To: [email protected]
> Subject: Re: st: Standard error of a ratio of two random variables
>
>
> For what it's worth, a simulation may give the most accurate answer:
>
> quietly ci X
> local mux=r(mean)
> local sex=r(se)
> quietly ci Y
> local muy=r(mean)
> local sey=r(se)
> corr X Y
> local rho=r(rho)
> matrix b= (`mux' \ `muy')
> matrix V= (1, `rho' \ `rho', 1)
> matrix sd =(`sex' \ `sey')
>
> clear
> set obs 100000
> drawnorm X Y, means(b) corr(V) sds(sd)
> gen ratio=X/Y
> sum ratio
>
> The SD of the variable -ratio- should be the SE of X/Y.
>
> Jeph
>
>
>
>
> Sergiy Radyakin wrote:
> > Dear All,
> >
> > I need to find a standard error of Z a ratio of two random
> > variables X and Y: Z=X/Y, where the means and SEs for X and Y are
> > known, as well as their corr coefficient. (In my particular case X and
>
> > Y are numbers of people that have such and such characteristics). I am
>
> > using delta-method according to [
> > http://www.math.umt.edu/patterson/549/Delta.pdf ] (see page 2(38)). I
> > then use svy:ratio to check my results and they don't match. I wonder
>
> > if I am doing something wrong, or is it any kind of precision-related
> > problem (the difference is about 7%, i.e. 1.7959 vs. 1.6795), or is my
>
> > check simply wrong and not applicable in this case.
> >
> > I would appreciate if someone could look into the code and let me
> > know why the results are different.
> >
> > Thank you,
> > Sergiy Radyakin
> >
> > Below is a do file that one can Ctrl+C/Ctrl+V to Stata's command line:
> >
> > **** BEGIN OF RV_RATIO.DO ****
> > sysuse auto, clear
> > generate byte www=1
> > svyset [pw=www]
> >
> > capture program drop st_error_of_ratio program define
> > st_error_of_ratio, rclass
> > syntax varlist(min=2 max=2)
> > svy: mean `varlist'
> > matrix B=e(b)
> > local mux=B[1,1]
> > local muy=B[1,2]
> > matrix V=e(V)
> > local sigma2x=V[1,1]
> > local sigma2y=V[2,2]
> > local sigmax_sigmay=V[1,2]
> > svyset
> > corr `varlist' [aw`=r(wexp)']
> > local rho=r(rho)
> > return scalar se = sqrt((`mux')^2*`sigma2y'/(`muy')^4 +
> > `sigma2x'/(`muy')^2 - 2*`mux'/(`muy')^3*`rho'*`sigmax_sigmay')
> > end
> > st_error_of_ratio price length
> > display as text "Estimated SE=" as result r(se)
> > svy: ratio price / length
> > **** END OF RV_RATIO.DO ****
> > *
> > * For searches and help try:
> > * http://www.stata.com/help.cgi?search
> > * http://www.stata.com/support/statalist/faq
> > * http://www.ats.ucla.edu/stat/stata/
> >
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
>
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
>
--
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
*
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